A car is traveling on a straight road. For $0 \leq t \leq 24$ seconds, the car's velocity $v ( t )$, in meters per second, is modeled by the piecewise-linear function defined by the graph above.
(a) Find $\int _ { 0 } ^ { 24 } v ( t ) \, dt$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) \, dt$.
(b) For each of $v ^ { \prime } ( 4 )$ and $v ^ { \prime } ( 20 )$, find the value or explain why it does not exist. Indicate units of measure.
(c) Let $a ( t )$ be the car's acceleration at time $t$, in meters per second per second. For $0 < t < 24$, write a piecewise-defined function for $a ( t )$.
(d) Find the average rate of change of $v$ over the interval $8 \leq t \leq 20$. Does the Mean Value Theorem guarantee a value of $c$, for $8 < c < 20$, such that $v ^ { \prime } ( c )$ is equal to this average rate of change? Why or why not?

Traffic congestion constitutes a problem that afflicts thousands of Brazilian drivers every day. The graph illustrates the situation, representing, over a defined time interval, the variation in the speed of a vehicle during a traffic jam.
How many minutes did the vehicle remain stationary throughout the total time interval analyzed?
(A) 4
(B) 3
(C) 2
(D) 1
(E) 0

The graph below shows the distance traveled by a cyclist as a function of time.
Based on the graph, what is the average speed of the cyclist, in km/h?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

A particle starts from the origin at time $t = 0$ and moves along the positive $x$-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time $t = 5s$?
(1) $10 m$
(2) $9 m$
(3) $6 m$
(4) $3 m$

The speed verses time graph for a particle is shown in the figure. The distance travelled (in $m$) by the particle during the time interval $\mathrm { t } = 0$ to $\mathrm { t } = 5$ s will be $\_\_\_\_$

From the $v - t$ graph shown, the ratio of distance to displacement in 25 s of motion is: [Figure]
(1) 1
(2) $\frac{1}{2}$
(3) $\frac{5}{3}$
(4) $\frac{3}{5}$

A disc is rolling without slipping on a surface. The radius of the disc is $R$. At $t = 0$, the top most point on the disc is $A$ as shown in figure. When the disc completes half of its rotation, the displacement of point $A$ from its initial position is
(1) $2R$
(2) $R\sqrt{\left(\pi^2 + 4\right)}$
(3) $R\sqrt{\left(\pi^2 + 1\right)}$
(4) $2R\sqrt{\left(1 + 4\pi^2\right)}$

A robot cat starts from the origin on a number line and moves in the positive direction. Its movement method is as follows: With an 8-second cycle, it moves at a constant speed of 4 units per second for 6 seconds, then rests for 2 seconds. Continuing this way, the robot cat will reach the position with coordinate 116 on the number line after $\underbrace{(14)(15)}$ seconds from the start of movement.

Two light spots move on a straight track of length 120 cm. When they reach an endpoint, they reverse direction and continue moving. Initially, the two spots are at the two ends of the track moving toward each other. Light spot A and light spot B move at speeds of 5 cm/s and 10 cm/s respectively. Select the correct options.
(1) The position of the first meeting of the two light spots is 40 cm away from one of the endpoints
(2) The position of light spot A exhibits periodic behavior with a period of 24 seconds
(3) When light spot A returns to its starting point, light spot B is also at its starting point
(4) The second meeting of the two light spots occurs at one of the endpoints
(5) There are 3 different meeting positions for the two light spots on the track

In the graph below showing Fatih's time to reach work according to his departure time from home on a certain day, the graph representations between 07.00-08.00 and 08.00-09.00 are linear.
Fatih, who left home at some time between 08.00 and 09.00, would have arrived at work at the same time if he had left home exactly one hour earlier.
Accordingly, at what time did Fatih arrive at work?
A) 09.12 B) 09.15 C) 09.18 D) 09.21 E) 09.24

An athlete ran from point A in the direction of the arrow on a circular track divided into five equal-length sections at a constant high speed and reached point B in 60 seconds. Then this athlete ran from point B in the direction of the arrow at a constant low speed for 320 seconds.
Accordingly, what is the difference between the time it takes this athlete to run the entire track at low speed and the time it takes to run the entire track at high speed in seconds?
A) 80 B) 90 C) 100 D) 110 E) 120